# First Order Theories of Some Lattices of Open Sets

## TL;DR

This paper investigates the logical complexity of the lattice of open sets in various topological spaces, showing equivalences to second order arithmetic and undecidability results for computable spaces.

## Contribution

It establishes the first order theories of these lattices as equivalent to second order arithmetic or undecidable, depending on the space.

## Key findings

- Lattice of open sets in some spaces is m-equivalent to second order arithmetic.
- For many computable spaces, the theory of effectively open sets is undecidable.
- In spaces like ^n, the theory is m-equivalent to first order arithmetic.

## Abstract

We show that the first order theory of the lattice of open sets in some natural topological spaces is $m$-equivalent to second order arithmetic. We also show that for many natural computable metric spaces and computable domains the first order theory of the lattice of effectively open sets is undecidable. Moreover, for several important spaces (e.g., $\mathbb{R}^n$, $n\geq1$, and the domain $P\omega$) this theory is $m$-equivalent to first order arithmetic.